Suppose that salaries for recent graduates of one university have a mean of $26,400 with a standard deviation of $1200.  Using Chebyshev's Theorem, what is the minimum percentage of recent graduates who have salaries between $22,800 and $30,000?  Round your answer to one decimal place.

To solve this problem using **Chebyshev's Theorem**, we first need to understand the theorem and then apply it to the given salary data.

### Chebyshev's Theorem:
Chebyshev's Theorem states that for any distribution (not necessarily normal), the proportion of observations that lie within $ k $ standard deviations from the mean is at least:
\[
1 - \frac{1}{k^2}
\]
where $ k $ is the number of standard deviations from the mean.

### Step 1: Find the mean and standard deviation
The mean salary is $ \mu = 26,400 $, and the standard deviation is $ \sigma = 1,200 $.

### Step 2: Determine the boundaries in terms of standard deviations
We are asked to find the percentage of recent graduates who have salaries between \$22,800 and \$30,000. We will determine how many standard deviations these boundaries are from the mean.

- Lower boundary: $ 22,800 $
  \[
  k_1 = \frac{26,400 - 22,800}{1,200} = \frac{3,600}{1,200} = 3
  \]
- Upper boundary: $ 30,000 $
  \[
  k_2 = \frac{30,000 - 26,400}{1,200} = \frac{3,600}{1,200} = 3
  \]

So both boundaries are 3 standard deviations away from the mean.

### Step 3: Apply Chebyshev's Theorem
Using Chebyshev's Theorem, we can find the minimum percentage of data that lies within 3 standard deviations of the mean:
\[
1 - \frac{1}{k^2} = 1 - \frac{1}{3^2} = 1 - \frac{1}{9} = 1 - 0.1111 = 0.8889
\]
This gives us 88.89%, or rounded to one decimal place, **88.9%**.

### Conclusion:
According to Chebyshev's Theorem, at least **88.9%** of the recent graduates have salaries between \$22,800 and \$30,000.

Would you like further details or have any questions?

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**Relative Questions:**
1. How does Chebyshev's Theorem differ from the Empirical Rule?
2. Can Chebyshev’s Theorem be applied to normal distributions as well?
3. How would the result change if the standard deviation increased?
4. What if we wanted to calculate the percentage within 2 standard deviations instead of 3?
5. How can Chebyshev's Theorem be useful in real-world data analysis?

**Tip:** Chebyshev's Theorem is useful for any dataset, especially when we don't know if the data follows a normal distribution.

Suppose that salaries for recent graduates of one university have a mean of $26,400 with a standard deviation of $1200. Using Chebyshev's Theorem, what is the minimum percentage of recent graduates who have salaries between $22,800 and $30,000? Round your answer to one decimal place.

Find out how to calculate the minimum percentage of recent graduates' salaries between $22,800 and $30,000 using Chebyshev's Theorem. With a mean of $26,400 and a standard deviation of $1,200, this detailed step-by-step guide applies key statistical concepts. Ideal for students in Grades 10-12.

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